Distribution function of plasma particles in axial magnetic and radial electric fieldsfor the transverse injection of neutral gas
The distribution function of particles in plasma which is created in crossed axial magnetic and radial electric fields by ionization of gas is obtained. It is assumed that the neutral gas before its ionization rotates with a constant angular velocity and the gas particle velocity distribution func...
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nasplib_isofts_kiev_ua-123456789-1153282025-02-23T18:13:51Z Distribution function of plasma particles in axial magnetic and radial electric fieldsfor the transverse injection of neutral gas Функция распределения частиц плазмы в осевом магнитном и радиальном электрическом полях при поперечной инжекции нейтрального газа Функція розподілу частинок плазми в осьовому магнітному і радіальному електричному полях при поперечній інжекції нейтрального газу Chibisov, D.V. Basic plasma The distribution function of particles in plasma which is created in crossed axial magnetic and radial electric fields by ionization of gas is obtained. It is assumed that the neutral gas before its ionization rotates with a constant angular velocity and the gas particle velocity distribution function in rotating frame is Maxwellian. Produced plasma particles move in crossed fields without collisions. The obtained distribution function is written in the coordinates of the guiding center. The expressions for the distribution function in the various special cases are also obtained. Получена функция распределения частиц в плазме, которая создаётся в скрещённых осевом магнитном и радиальном электрическом полях. Предполагается, что нейтральный газ перед ионизацией вращается с по- стоянной угловой скоростью, а функция распределения частиц газа по скоростям во вращающейся системе отсчёта является максвелловской. Образовавшиеся частицы плазмы движутся в скрещённых полях без столкновений. Полученная функция распределения записана в координатах ведущего центра. Получены также выражения для функции распределения в различных частных случаях. Отримано функцію розподілу частинок у плазмі, яка утворюється в схрещених осьовому магнітному і радіальному електричному полях. Передбачається, що нейтральний газ перед іонізацією обертається зі ста- лою кутовою швидкістю, а функція розподілу частинок газу за швидкостями в обертовій системі відліку є максвеллівською. Утворині частинки плазми рухаються в схрещених полях без зіткнень. Отримана функція розподілу записана в координатах ведучого центру. Отримано також вирази для функції розподілу в різних окремих випадках. 2016 Article Distribution function of plasma particles in axial magnetic and radial electric fieldsfor the transverse injection of neutral gas / D.V. Chibisov // Вопросы атомной науки и техники. — 2016. — № 6. — С. 104-107. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 52.27.Jt https://nasplib.isofts.kiev.ua/handle/123456789/115328 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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| topic |
Basic plasma Basic plasma |
| spellingShingle |
Basic plasma Basic plasma Chibisov, D.V. Distribution function of plasma particles in axial magnetic and radial electric fieldsfor the transverse injection of neutral gas Вопросы атомной науки и техники |
| description |
The distribution function of particles in plasma which is created in crossed axial magnetic and radial electric
fields by ionization of gas is obtained. It is assumed that the neutral gas before its ionization rotates with a constant
angular velocity and the gas particle velocity distribution function in rotating frame is Maxwellian. Produced plasma
particles move in crossed fields without collisions. The obtained distribution function is written in the coordinates of
the guiding center. The expressions for the distribution function in the various special cases are also obtained. |
| format |
Article |
| author |
Chibisov, D.V. |
| author_facet |
Chibisov, D.V. |
| author_sort |
Chibisov, D.V. |
| title |
Distribution function of plasma particles in axial magnetic and radial electric fieldsfor the transverse injection of neutral gas |
| title_short |
Distribution function of plasma particles in axial magnetic and radial electric fieldsfor the transverse injection of neutral gas |
| title_full |
Distribution function of plasma particles in axial magnetic and radial electric fieldsfor the transverse injection of neutral gas |
| title_fullStr |
Distribution function of plasma particles in axial magnetic and radial electric fieldsfor the transverse injection of neutral gas |
| title_full_unstemmed |
Distribution function of plasma particles in axial magnetic and radial electric fieldsfor the transverse injection of neutral gas |
| title_sort |
distribution function of plasma particles in axial magnetic and radial electric fieldsfor the transverse injection of neutral gas |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2016 |
| topic_facet |
Basic plasma |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/115328 |
| citation_txt |
Distribution function of plasma particles in axial magnetic and radial electric fieldsfor the transverse injection of neutral gas / D.V. Chibisov // Вопросы атомной науки и техники. — 2016. — № 6. — С. 104-107. — Бібліогр.: 8 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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| first_indexed |
2025-11-24T06:13:07Z |
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2025-11-24T06:13:07Z |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2016. №6(106)
104 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2016, № 6. Series: Plasma Physics (22), p. 104-107.
DISTRIBUTION FUNCTION OF PLASMA PARTICLES IN AXIAL MAG-
NETIC AND RADIAL ELECTRIC FIELDSFOR THE TRANSVERSE IN-
JECTION OF NEUTRAL GAS
D.V. Chibisov
V.N. Karazin Kharkiv National University, Department of Physics, Kharkov, Ukraine
E-mail: chibisovdm@mail.ru
The distribution function of particles in plasma which is created in crossed axial magnetic and radial electric
fields by ionization of gas is obtained. It is assumed that the neutral gas before its ionization rotates with a constant
angular velocity and the gas particle velocity distribution function in rotating frame is Maxwellian. Produced plasma
particles move in crossed fields without collisions. The obtained distribution function is written in the coordinates of
the guiding center. The expressions for the distribution function in the various special cases are also obtained.
PACS: 52.27.Jt
INTRODUCTION
Particle distribution function of cylindrically sym-
metric plasma in crossed axial magnetic and radial elec-
tric fields depends on the conditions of its creation. In
[1,2] the distribution function of plasma created as a
result of ionization of a cold gas was considered:
zez veYvF 0,, , (1)
where Y is the Heaviside step function, is the Dirac
delta function, 0 is the potential on the external elec-
trode (anode), /e rcE Br . Variables ,
and zv are the energy of the transverse motion, the gen-
eralized angular momentum and velocity along the
magnetic field respectively. The radial profile of the
electric field potential was assumed parabolic
2
0 )()( arr , which ensured the independence of
the rate of angular rotation of the particle on the radius.
A distinctive feature of that plasma was a strong radial
electric field produced by uncompensated charge of
electrons, so that the ions move radially in very elongat-
ed orbits and azimuthal precession with frequency
2ci where ci is the ion cyclotron frequency. That
model was applied in a Penning discharge with a low-
density plasma ( 0n ~ 39cм10 ) in the analysis of plasma
stability in particular in the study of low frequency os-
cillations.
In [3] the expression for the plasma particle distribu-
tion functions in crossed axial magnetic and radial elec-
tric fields, where the thermal motion of atoms before
their ionization taken into account, was obtained:
R
v
I
v
n
VRf
T
c
T
z 2
0
03
0
21
0
0
2
,,
2
0
2
2
0
22
2
0
22
222
exp
T
z
T
c
T v
v
vv
R
Y R a , (2)
where and R are variables of the guiding center of
particle species respectively Larmor radius and radi-
al coordinate of the center of the Larmor circle, 0n and
0Tv are the density and the thermal velocity of the par-
ticles before to their appearance in crossed fields.
0I x is the Bessel function of imaginary argument,
is the angular velocity of the drift motion of parti-
cles in crossed fields:
22
08
11
2 am
e
c
c
. (3)
This distribution function can be used to analyze the
stability of plasma created in crossed fields of vapors of
substance that have high boiling point. The application
of that plasma, created by the reflective discharge in
order to separate elements and isotopes was discussed in
[4-6]. The usual analysis of the stability of plasma with
a known function of the distribution can also be sup-
plemented by searching the optimum particle distribu-
tion function of vapor of working substance, whereby
the efficiency of the separation of elements would be
enhanced. In this paper, the function of the plasma par-
ticle distribution was obtained under conditions when a
neutral gas (or vapor of working substance) before ioni-
zation rotates with a certain angular velocity, taking into
account the thermal spread of velocities of the gas mol-
ecules in the rotating frame of reference. Choosing an
appropriate gas angular rotation velocity depending on
the ion cyclotron frequency of a particular element, or
the velocity of the drift motion of plasma particles in
crossed fields makes it possible to control the particle
distribution function of plasma, thereby regulate the
plasma wave processes, as well as affect on the process
of separation elements.
1. GENERAL EXPRESSION FOR THE
DISTRIBUTION FUNCTION
Now we obtain the particle distribution function of
plasma in the crossed fields assuming that the gas ro-
tates at a constant angular velocity before its ionization.
Here we use the same considerations as in [3] where gas
initially was in rest.
Suppose that cylindrically symmetric collisionless
plasma is placed in crossed longitudinal magnetic and
radial electric fields. In the radial direction plasma is
limited by metal electrode (anode) with the radius a
and which has a positive potential relative to the axis, so
that the electric field is directed into the plasma. We
assume that the potential in plasma has a quadratic
ISSN 1562-6016. ВАНТ. 2016. №6(106) 105
dependence on the radius
2
0( ) ( )r r a so that the
rotational velocity of the particles is not depends on the
radius. This potential distribution occurs in negatively
charged plasma with uniform radial distribution of the
electron density. Along the magnetic field the plasma is
considered to be unlimited. The equilibrium distribution
function of plasma particles of a species (ion or elec-
tron) should depends on variables , , zv , where
rvm
rm
re
vm c
2
,
2
22
, (4)
2 2 2
rv v v . Suppose that the particle of species
appeared in the crossed fields with the initial values of
the radial coordinate 0r and velocity components
0 0 0, ,r zv v v . The probability that, moving in crossed
fields, the particle will be in the phase volume
zd d dv is equal to:
2
())(
2
(
2
0
0
2
0
crm
re
vm
dp
0 0 0) ( )z z zm v r v v d d dv . (5)
This form of dependence of the probability density is a
consequence of the laws of conservation of energy and
angular momentum of a particle. If, before the appear-
ance of the particles in crossed fields, they have a cer-
tain distribution of the coordinates and velocities
0 0 0 0( , , , )r zf r v v v , then the probability that the parti-
cle will be found in the phase volume
0 0 0 0 0z r zd d dv r dr dv dv dv is equal to:
0 0 0 0 0 0 0 0( , , )z r zdP f r v v dpr dr dv dv dv . (6)
In order to obtain the particle distribution function, the
expression (6) should be integrated over the variables
0 0 0 0, , ,r zr v v v :
2
0
0 0 0 0 0( , ) ( , , , ) ( ( ))
2
r z
m v
F f r v v v e r
2
0
0 0 0 0 0 0 0 0( ) ( )
2
c
z z r z
m r
m v r v v r dr dv dv dv
. (7)
Integration over variables 0 0 0, ,r zv v v gives
0 0 0 0 02
0 0
1
( , ) ( , , , )r z
r
F f r v v v dr
m v
, (8)
where
2
0 0
0
1
(2 )
2
cv m r
m r
,
1 2
2
2 2
0 *
* 0
1
4 2 4
2
r c c
c
v
m r
1 2
2
2 2
* 0
2
2 2
*
4 2
1
4 2 4
c c
c c
m r
,
* 2c c is the modified cyclotron frequen-
cy in the crossed fields. Now we replace in integral (8)
the variable 0r by as
2
0 2
*
4 2
cos
c
c
r
m
1 2
2
2
2 2 2
* *
4 2 4c
c cm m
. (9)
It gives
0 0 0 02
0*
1
( , ) ( , , , )r z
c
F f r v v v
m
0aY r r d . (10)
Suppose that the distribution function
0 0 0 0( , , , )r zf r v v v is given by:
2
0 0
0 0 0 0 3 2 3 2
0 0
( , , , ) exp
(2 ) 2
r
r z
T T
n v
f r v v v
v v
2
2
0 0 0 0
02 2
0 0
( )
2 2
z
a
T T
v r v
Y r r
v v
, (11)
that corresponds initially rotating gas with angular ve-
locity 0 . Then, taken into account (9), we obtain
00
2 3 2 3 2 2
00 0 0
1
( , ) exp
(2 )c T T T
n
F
m v m v m v
2
0
0 02 2
0 0
1
2 2
z
c c
T T
v
v v
1 2
2
2
2 2 2 2
* * *
4 2 4 2 4
cos
c c
c c cm m m
drrY a 0 . (12)
In the frame of reference rotating with the angular ve-
locity particle of species to perform a circular
motion like Larmor rotation in the magnetic field. The
role of the cyclotron frequency in this case has a modi-
fied cyclotron frequency 2c c and the
Larmor radius as well as the radial coordinate of the
center of the Larmor circle R (variables of the guiding
center of the particle) related to the variables and
by the following relations [7]
2
*
1 2
c
m
, (13)
2
*
1 2
c
c
R
m
. (14)
Note that and R are also integrals of motion. Dis-
tribution function in the variables of the guiding center
will have the form
0
2 3 2 3
0
1
( , )
(2 )c T
n
F
m v
2 22 2
0 02 2
0 0 0
1 1
exp
2 2
c
T T
R
v v
106 ISSN 1562-6016. ВАНТ. 2016. №6(106)
2
0
0 0 2
0
cos
2
z
c
T
v
R
v
0rRY . (15)
Integrating (15) over , we obtain
0
0 3
0
, ,
2
z
T
n
f R v
v
0 0 0 2
0
c
T
R
I
v
22
02
0
1
exp
2
c
Tv
2
22 0
0 02 2
0 0
1
2 2
z
T T
v
R Y R r
v v
. (16)
Expression (16) is the desired distribution function of
particles in crossed fields for initially rotated gas with
an arbitrary relationship between the energy of motion
of particles in an electric field and their energy of ther-
mal motion. Note, that in the limiting case
0 0 ex-
pression (16) reduces to (2).
2. SPECIAL CASES FOR THE
DISTRIBUTION FUNCTION
Now we consider the limiting cases of the distribu-
tion function for the different limit ratios of the thermal
velocity 0Tv and the 0 0cR
value.
Assume first that 0Tv ~0, while the anode potential 0
is high enough so that most of the volume of plasma
(except the paraxial region) the inequality
2
000 Tc vR (17)
holds. Then, for the Bessel functions, we can use the
asymptotic form for large values of the argument
xI0 ~ xex 2 . In this case the distribution function
(10) has the form
0
0 2
0
, ,
2
z
T
n
f R v
v
0 0
1
c R
2
0
2
2
0
2
00
22
exp
T
z
T
c
v
v
v
R
0Y R r . (18)
It follows from Eq. (18), that the main contribution to
the equilibrium distribution function gives the particles,
which satisfy to condition:
000 Tc vR . (19)
Inequality (19) determines the approximation of a
strong radial electric field when the thermal velocity of
the particles before their appearance in crossed fields is
much less than the velocity of the particle drift in
crossed fields RVT 00 . In the limiting case
0 0Tv the distribution function (18) reduces into
-function:
000 , cRRf . (20)
Similar ion distribution function in the form of
-function (1) was considered in [1,2], where the prob-
lem of the excitation of the ion cyclotron instability of
the plasma in crossed B and rE fields was solved.
Assume now that the inequality opposite to (17)
2
000 Tc vR (21)
holds. This inequality corresponds to the case of a weak
electric field. In this case, the distribution function (16)
takes the form:
3
0
21
0
0
2
,,
T
z
v
n
vRf
2
0
2
2
0
22
0
2
0
22
0
222
exp
T
z
T
c
T v
v
vv
R
Y R a . (22)
Note that despite the uniform ionization along the radi-
us, the distribution function (22) is Gaussian on radial
coordinate of the guiding center.
Now we will obtain the distribution function for the
particular values of angular velocity
0 . Suppose in
(16)
0 , i.e. angular velocity rotation of the gas
coincides with the angular velocity of the drift motion
of particles in crossed fields. Then the distribution func-
tion has the form
3
0
21
0
0
2
,,
T
z
v
n
vRf
2
0
2
0
2
0
22
*
22
exp
T
z
T
c
v
v
v
Y R a . (23)
It is obvious that distribution function (23) not depends
on R and plasma is homogeneous. This is true for
arbitrary ratios of the thermal velocity and the velocity
of the drift motion of the guiding center.
Assume now, that
0 c . Then the distribution
function (16) equals
0
0 03 2
0 0
, ,
2
z c
T T
n R
f R v I
v v
2
0
2
022
2
0
22
2
0 22
1
2
1
exp
T
z
c
TT v
v
R
vv
0Y R r . (24)
As can be seen the expression (24) is obtained from (16)
by interchange of variables R . The distribution
function (24) corresponds to the azimuthal flow compo-
nent , rotating with angular velocity
c ,
however written in the laboratory frame. As shown in
[8], the transition to a frame of reference rotating with
angular velocity c leads to interchange of vari-
ables R in distribution function and thus in a
new frame of reference, we again obtain the expression
(16).
ISSN 1562-6016. ВАНТ. 2016. №6(106) 107
Now we consider the case 0 c .
Then the distribution function takes the form
2 2 2
0 * 0
0 1 2 2 23
0 00
, , exp
2 22
c z
z
T TT
n R v
f R v
v vv
Y R a . (25)
This expression coincides with (23) up to the replace-
ment R , and corresponds to particles encircling
plasma axis.
CONCLUSIONS
The expression for the distribution function of plas-
ma particles in crossed longitudinal magnetic and radial
electric field using the probabilistic approach is ob-
tained. It is assumed that gas up to the ionization rotates
with angular velocity
0 . This expression takes into
account non-zero initial velocity of the atoms in rotating
frame. The distribution function includes the product of
the modified Bessel function and exponential (16)
whose arguments are the coordinates of the guiding cen-
ter.
From the general expression for the distribution
function the limiting expressions in the cases of strong
(18) and weak (22) radial electric field was obtained.
These expressions are consistent with the previously
obtained expressions.
The expressions for the distribution function for par-
ticular values of the angular velocity
0 where ob-
tained, in particular:
1) When the equality
0 satisfied the distribution
function (23) not depends on R and plasma is homo-
geneous;
2) When the equality
0 c satisfied the distribu-
tion function (24) corresponds to (16) with interchange
of variables R .
3) When the equality 0 c satisfied the
distribution function (25) corresponds to (23) with in-
terchange of variables R .
REFERENCES
1. V.G. Dem’yanov, Yu.N. Eliseev, Yu.A. Kirochkin, et
al. Equilibrium and non-local ion cyclotron instability of
plasma in crossed longitudinal magnetic and strong ra-
dial electric fields // Fiz. Plazmy. 1988, v. 14, № 10,
p. 840-850 (in Russian).
2. Yu.N. Yeliseyev. Nonlocal theory of the spectra of
modified ion cyclotron oscillations in a charged plasma
produced by gas ionization // Plasma Phys. Rep. 2006,
v. 32, № 11, p. 927-936.
3. D.V. Chibisov. The distribution function of plasma
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Article received 22.09.2016
ФУНКЦИЯ РАСПРЕДЕЛЕНИЯ ЧАСТИЦ ПЛАЗМЫ В ОСЕВОМ МАГНИТНОМ И РАДИАЛЬНОМ
ЭЛЕКТРИЧЕСКОМ ПОЛЯХ ПРИ ПОПЕРЕЧНОЙ ИНЖЕКЦИИ НЕЙТРАЛЬНОГО ГАЗА
Д.В. Чибисов
Получена функция распределения частиц в плазме, которая создаётся в скрещённых осевом магнитном и
радиальном электрическом полях. Предполагается, что нейтральный газ перед ионизацией вращается с по-
стоянной угловой скоростью, а функция распределения частиц газа по скоростям во вращающейся системе
отсчёта является максвелловской. Образовавшиеся частицы плазмы движутся в скрещённых полях без
столкновений. Полученная функция распределения записана в координатах ведущего центра. Получены
также выражения для функции распределения в различных частных случаях.
ФУНКЦІЯ РОЗПОДІЛУ ЧАСТИНОК ПЛАЗМИ В ОСЬОВОМУ МАГНІТНОМУ І РАДІАЛЬНОМУ
ЕЛЕКТРИЧНОМУ ПОЛЯХ ПРИ ПОПЕРЕЧНІЙ ІНЖЕКЦІЇ НЕЙТРАЛЬНОГО ГАЗУ
Д.В. Чібісов
Отримано функцію розподілу частинок у плазмі, яка утворюється в схрещених осьовому магнітному і
радіальному електричному полях. Передбачається, що нейтральний газ перед іонізацією обертається зі ста-
лою кутовою швидкістю, а функція розподілу частинок газу за швидкостями в обертовій системі відліку є
максвеллівською. Утворині частинки плазми рухаються в схрещених полях без зіткнень. Отримана функція
розподілу записана в координатах ведучого центру. Отримано також вирази для функції розподілу в різних
окремих випадках.
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